A Two-Way Parallel Slime Mold Algorithm by Flow and Distance for the Travelling Salesman Problem
A Two-Way Parallel Slime Mold Algorithm by Flow and Distance for the Travelling Salesman Problem
Liu, Meijiao;Li, Yanhui;Huo, Qi;Li, Ang;Zhu, Mingchao;Qu, Nan;Chen, Liheng;Xia, Mingyi
2020-09-05 00:00:00
applied sciences Article A Two-Way Parallel Slime Mold Algorithm by Flow and Distance for the Travelling Salesman Problem 1 , 2 1 1 1 , 2 1 , 3 , 1 Meijiao Liu , Yanhui Li , Qi Huo , Ang Li , Mingchao Zhu *, Nan Qu *, Liheng Chen and Mingyi Xia Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, Changchun 130033, China; liumeijiao17@mails.ucas.ac.cn (M.L.); liyanhui@ciomp.ac.cn (Y.L.); huoqi@ciomp.ac.cn (Q.H.); liang@ciomp.ac.cn (A.L.); chenliheng@ciomp.ac.cn (L.C.); xiamingyi@ciomp.ac.cn (M.X.) School of Optoelectronics, University of Chinese Academy of Sciences, Beijing 100049, China College of Resources and Environmental Sciences, Jilin Agricultural University, Changchun 130118, China * Correspondence: zhumingchao@ciomp.ac.cn or mingchaozhu@gmail.com (M.Z.); qunan@jlau.edu.cn (N.Q.) Received: 9 July 2020; Accepted: 3 September 2020; Published: 5 September 2020 Featured Application: The design of this article can be applied to Travelling Salesman Problem and related problems. Abstract: In order to solve the problem of poor local optimization of the Slime Mold Algorithm (SMA) in the Travelling Salesman Problem (TSP), a Two-way Parallel Slime Mold Algorithm by Flow and Distance (TPSMA) is proposed in this paper. Firstly, the flow between each path point is calculated by the “critical pipeline and critical culture” model of SMA; then, according to the two indexes of flow and distance, the set of path points to be selected is obtained; finally, the optimization principle with a flow index is improved with two indexes of flow and distance and added random strategy. Hence, a two-way parallel optimization method is realized and the local optimal problem is solved eectively. Through the simulation of Traveling Salesman Problem Library (TSPLIB) on ulysses16, city31, eil51, gr96, and bier127, the results of TPSMA were improved by 24.56, 36.10, 41.88, 49.83, and 52.93%, respectively, compared to SMA. Furthermore, the number of path points is more and the optimization ability of TPSMA is better. At the same time, TPSMA is closer to the current optimal result than other algorithms by multiple sets of tests, and its time complexity is obviously better than others. Therefore, the superiority of TPSMA is adequately proven. Keywords: Slime Mold Algorithm; two-way parallel optimization; flow and distance; Travelling Salesman Problem 1. Introduction The Travelling Salesman Problem (TSP) [1,2] is a classical problem in Non-Deterministic Polynomial problems, and has important practical significance in road network planning, workshop dispatch, and so on. With the development of the heuristic bionic algorithms and their good eect on solving problems, TSP has been solved by various intelligent algorithms such as Genetic Algorithm (GA) [3,4], Particle Swarm Optimization (PSO) [5,6], Ant Colony Optimization (ACO) [7,8], the classic heuristic Lin-Kernighan [9,10] and Lin-Kernighan-Helsgaun Solver (LKH) [11,12], etc. PSO is simple, but its eect is not good. The eect of GA is general and the algorithm is complex. ACO has a good eect, but its convergence speed is slow. For Lin-Kernighan, its complexity increases exponentially with the increase in the number of path points, so it takes too long to obtain results. LKH is an improved algorithm on the basis of Lin-Kernighan. Although the optimization method is the best so far in TSP, the time complexity is still too large to optimize quickly. Appl. Sci. 2020, 10, 6180; doi:10.3390/app10186180 www.mdpi.com/journal/applsci Appl. Sci. 2020, 10, x FOR PEER REVIEW 2 of 20 Appl. Sci. 2020, 10, 6180 2 of 15 improved algorithm on the basis of Lin-Kernighan. Although the optimization method is the best so far in TSP, the time complexity is still too large to optimize quickly. The microorganism called slime mold [13] has an evolutionary history of 100 million years. The microorganism called slime mold [13] has an evolutionary history of 100 million years. It It can eventually find a high-quality foraging path by avoiding external obstacles in a network can eventually find a high-quality foraging path by avoiding external obstacles in a network structure. The unique and ecient foraging process provides a new inspiration for path optimization. structure. The unique and efficient foraging process provides a new inspiration for path optimization. Therefore, scientists learned from the intelligent life and obtained a new algorithm called Slime Mold Therefore, scientists learned from the intelligent life and obtained a new algorithm called Slime Mold Algorithm (SMA) [14]. With the unique searching ability, slime molds can eciently find a high-quality Algorithm (SMA) [14]. With the unique searching ability, slime molds can efficiently find a high- way to the food in maze experiments. As shown in Figure 1, based on the slime mold’s foraging quality way to the food in maze experiments. As shown in Figure 1, based on the slime mold’s behavior, the scientists thought of the food sources as the cities of a country. It was found that the foraging behavior, the scientists thought of the food sources as the cities of a country. It was found foraging path network of slime molds is highly similar to the actual road network designed in the that the foraging path network of slime molds is highly similar to the actual road network designed country [15–18]. Thus, foraging behavior of slime mold has great significance to be explored for path in the country [15–18]. Thus, foraging behavior of slime mold has great significance to be explored optimization [19–23]. for path optimization [19–23]. Figure 1. Foraging network of slime mold and road network. Figure 1. Foraging network of slime mold and road network. In 2000, Japanese scientist Nakagaki [14,24–26] discovered that the slime mold can walk through In 2000, Japanese scientist Nakagaki [14,24–26] discovered that the slime mold can walk through the maze. The researchers placed the slime molds in a maze and dropped food at the entrance and exit the maze. The researchers placed the slime molds in a maze and dropped food at the entrance and of the maze in a petri dish. After a period of time, slime molds formed a feeding route. It was obtained exit of the maze in a petri dish. After a period of time, slime molds formed a feeding route. It was as the solution to solve a complex maze problem. In 2007, Tero [27,28] came up with the model of obtained as the solution to solve a complex maze problem. In 2007, Tero [27,28] came up with the “pipeline culture,” which mainly used Poisson Theorem and Kirchho’s Laws to realize the pipeline model of “pipeline culture,” which mainly used Poisson Theorem and Kirchhoff’s Laws to realize the mechanism of flow and conductivity. This model can be equivalent to the foraging behavior of slime pipeline mechanism of flow and conductivity. This model can be equivalent to the foraging behavior molds. Afterwards, Gunji et al. [29,30] applied the model to the description of networks in a cellular of slime molds. Afterwards, Gunji et al. [29,30] applied the model to the description of networks in a model. In China, Southwest University has studied the Slime Mold Algorithm combined with the cellular model. In China, Southwest University has studied the Slime Mold Algorithm combined with pheromones of the Ant Colony Optimization to solve the classic TSP and multi-object TSP [31–34]. the pheromones of the Ant Colony Optimization to solve the classic TSP and multi-object TSP [31– Compared with other algorithms, SMA has high-eciency optimization ability, especially in solving 34]. Compared with other algorithms, SMA has high-efficiency optimization ability, especially in the complex path problem, which includes a large number of points and complex distribution [35–37]. solving the complex path problem, which includes a large number of points and complex distribution The way of SMA has made a new method to solve TSP. At the same time, the convergence of SMA [35–37]. The way of SMA has made a new method to solve TSP. At the same time, the convergence is fast, due to fewer iterations. However, SMA has been researched more recently than the others, of SMA is fast, due to fewer iterations. However, SMA has been researched more recently than the and there is still a lot of room to investigate and improve [1,2,27,28,33,34]. In summary, the bottlenecks others, and there is still a lot of room to investigate and improve [1,2,27,28,33,34]. In summary, the besides its advantages in TSP are as follows: bottlenecks besides its advantages in TSP are as follows: Due to the high similarity of some flow values, SMA cannot make a suitable choice. If points are • Due to the high similarity of some flow values, SMA cannot make a suitable choice. If points are selected only by flow, the ability will have a great limitation of global optimization. selected only by flow, the ability will have a great limitation of global optimization. SMA has no randomness and the selected points and the points to be selected have strong • SMA has no randomness and the selected points and the points to be selected have strong correlations. Therefore, SMA has low flexibility and weak robustness. correlations. Therefore, SMA has low flexibility and weak robustness. The Two-way Parallel Slime Mold Algorithm (TPSMA) by flow and distance for TSP is proposed The Two-way Parallel Slime Mold Algorithm (TPSMA) by flow and distance for TSP is proposed in this paper. TPSMA involves two indicators of flow and distance for path selection and adds in this paper. TPSMA involves two indicators of flow and distance for path selection and adds random random factors. TPSMA will improve the quality of SMA for solving TSP and achieve the following factors. TPSMA will improve the quality of SMA for solving TSP and achieve the following advantages: advantages: The selection rule combines two indicators of flow and distance, which makes SMA not only rely • The selection rule combines two indicators of flow and distance, which makes SMA not only on the flow. It can better improve global optimization ability and prevent the algorithm from rely on the flow. It can better improve global optimization ability and prevent the algorithm falling into local optimum. from falling into local optimum. The proposed TPSMA adds random factors to increase the diversity of path choices and improve • The proposed TPSMA adds random factors to increase the diversity of path choices and improve the robustness of the algorithm. the robustness of the algorithm. Appl. Sci. 2020, 10, 6180 3 of 15 Appl. Sci. 2020, 10, x FOR PEER REVIEW 3 of 20 The structure of the rest of the article is: the second part describes the basic principle of SMA; The structure of the rest of the article is: the second part describes the basic principle of SMA; the third part describes the design of TPSMA, including a specific, improved strategy; the fourth part, the third part describes the design of TPSMA, including a specific, improved strategy; the fourth part, simulation and analysis; the fifth part, conclusions. simulation and analysis; the fifth part, conclusions. 2. Sl 2. Slime ime Mo Mold ld Al Algorithm gorithm According to the foraging behavior of slime mold, the basic idea of SMA is: the slime molds According to the foraging behavior of slime mold, the basic idea of SMA is: the slime molds expand expand to to ev every ery dir direc ection tion and andstr stre etch tch themselves themselves toto cover the sur cover the surroundings. roundings. Then, Then, the the slime slim mold e mold will shrink back to the direction without food or far away from food, and they will continue to expand will shrink back to the direction without food or far away from food, and they will continue to expand in the in the di dir rect ection ion of foo of food. d. TThat hat isis, , ifif they they fefeel el ththat at ththe e foo food d is is abun abundant, dant, the they y wiwill ll contin continue ue to exp to expand; and; if if they feel that the food is scarce, they will shrink and return. After a period of time, the slime molds they feel that the food is scarce, they will shrink and return. After a period of time, the slime molds wil will l fo form rm a p a path, ath, llike ike a pip a pipeline, eline, and and fin find d the the shor shortest test rou route. te. The “pipeline culture” model can be abstracted by slime mold foraging. The basic principle of The “pipeline culture” model can be abstracted by slime mold foraging. The basic principle of SMA SMA iis s [ [20– 20–22 22,3 ,31 1–3 –34 4] ]:: f firstly irstly, pi , pipeline peline pa paths ths a are re bu built ilin t in a all ll d directions irections and and form form a network a netwobetween rk between the food sources by imitating foraging behavior of slime mold; then, based on the length of each path, the food sources by imitating foraging behavior of slime mold; then, based on the length of each path, the w the width idth o off pipeline, pipeline,and andthe the obstacles obstacleon s on p path, ath foraging , foraging paths paths can cbe an b obtained; e obtained thir ; dly third , under ly, und theer e the ect of iteration, the stable distribution of flow will be formed through a period of dynamic transformation; effect of iteration, the stable distribution of flow will be formed through a period of dynamic tran finally sfo,ra m path ation is ; fgenerated inally, a pfr ath om isthe gener start ate position d from the to the star end t po that sition hasto the the en food. d The that h schematic as the food diagram . The of SMA is shown in Figure 2. According to the slime mold algorithm in solving TSP, we obtain the schematic diagram of SMA is shown in Figure 2. According to the slime mold algorithm in solving TSP, w “pipeline e obt cultur ain the e” model “pipelin [2e cu ,31–37 lture ], and ” mo the del specific [2,31–37] algorithm , and the specif is as follows: ic algorithm is as follows: Figure 2. Schematic diagram of “pipeline culture” model of SMA. Figure 2. Schematic diagram of “pipeline culture” model of SMA. (1) Variables initialize. (1) Variables initialize. (2) Calculate the distance between each path point by distance formula. The distance between (2) Calculate the distance between each path point by distance formula. The distance between each path point is calculated by: each path point is calculated by: 1/2 2 2 1/2 L = x x y y (1) ij i j i j Lx=−x −y−y (1) () ( ) ij ( i j i j ) where x , x represent the abscissa of i and j, and y , y represent the ordinate of i and j. L is defined as i j i j ij where x , x represent the abscissa of i and j , and y , y represent the ordinate of i and j . i j i j the distance between i and j, also called the pipeline length of i to j. L is defined as the distance between i and j , also called the pipeline length of i to j . ij (3) Each path point can be regarded as a node in the pipeline network, and we will select two path (3) Each path point can be regarded as a node in the pipeline network, and we will select two points as the entrance point and the exit point, respectively. Then, the pressure value of each path path points as the entrance point and the exit point, respectively. Then, the pressure value of each point is calculated according to Kirchho’s Laws. The formula is expressed as: path point is calculated according to Kirchhoff’s Laws. The formula is expressed as: I , f or j = 1 X > >−= If ,1 or j ij < 0 P P = I , f or j = 2 (2) > ij i j 0 L PP−= I ,2 for j= () (2) ij ij : 0 0 , otherwise ij 0s ,i otherwe where D represents the conductivity of pipeline between i and j. P and P are the pressure of point i ij i j where D represents the conductivity of pipeline between i and j . P and P are the pressure ij i j and j. The solution of conductivity needs to set a point pressure as the reference, and then calculate the of point i and j . The solution of conductivity needs to set a point pressure as the reference, and rest of pressure of each point. For example, setting p = 0 as the reference point of pressure. then calculate the rest of pressure of each point. For example, setting p =0 as the reference point of pressure. Appl. Sci. 2020, 10, 6180 4 of 15 (4) Q is defined as the flow value between i and j. Q needs to combine the relationship among ij ij the dierence of pressure P P , conductivity D , and distance L . The relation’s formula that i j ij ij calculates the flow of each path pipeline is: ij Q = P P (3) ij i j ij (5) D is the conductivity between i and j. The pipeline conductivity is required to be updated ij constantly. The calculation’s formula is as follows: dD Q ij ij = D (4) ij dt 1 + Q ij The iterative formula after deformation is: 0 1 Q (n) B C ij B C B C D (n + 1) = B D (n)C Dt + D (n) (5) ij ij ij @ A 1 + Q (n) ij (6) According to above process of (3) to (5), we complete the next cycle repeatedly until getting to the iteration termination condition. The stable value D and Q will be obtained by carrying out ij ij iteration, and the iterative condition of termination is defined as: D (n + 1) D (n) (6) ij ij (7) According to the final flow values, the next selected point is determined, and the path will be obtained by selecting the point of largest flow, from the starting point, one by one. After completing the selection of one point, the selected point P will become the current point i in the next point next selection process. The point i is recorded into L by Equation (8), and the path result L is finally best best obtained by SMA. The concrete formula is shown as: Q = maxfjQ j,jQ j,:::::: ,jQ jg (7) inext i1 i2 im L = fi , i ,:::::: , i g (8) best 1 2 n where Q represents the pipeline flow with the largest value from the current point i to the other inext points. At the same time, it is the flow of next path point selected. L is the path result by SMA. best 3. Two-Way Parallel Slime Mold Algorithm by Flow and Distance Due to the short development time, the research depth of SMA is insucient and many details of the model have to be explored. As far as the “pipeline cultivation” model of SMA, the path point selection is only based on the flow value, and there is only one path to be obtained, so it makes the optimization results limited. Moreover, SMA cannot jump out of the local optimum, especially in the complex situation of points. Therefore, TPSMA is designed by using two-way parallel optimization on flow and distance in this paper. As shown in Figure 3, it uses two reference indicators of flow and distance to search for next point, instead of the original principle, which is selecting the next point only by flow. At the same time, a random factor is added when points are selected by the flow and distance, to increase the diversity of result. Appl. Sci. 2020, 10, x FOR PEER REVIEW 5 of 20 random factor is added to achieve the selection of the next point when we select the points in point Appl. Sci. 2020, 10, 6180 5 of 15 set. Finally, the optimized result of TPSMA will be obtained by iterating. The selection rule of the original algorithm Delete the point Select Qbest LQbest PQbest LQbetter judge Qbetter - Select Lbest rand set Prand Lbetter Delete the point The selection rule of the improved algorithm Figure Figure 3. 3. Schematic Schematicdiagram diagram of al of algorithm gorithm pri principle nciple by TPSMA. by TPSMA. The schematic diagram of TPSMA is illustrated in Figure 3. Firstly, the flow value between points According to the principle of TPSMA on flow and distance, the algorithm is applied to the is obtained by using the “pipeline culture” model of SMA. Secondly, the path point can be selected solution of TSP. As illustrated in Figure 4, the specific steps are as follows: by the flow and distance, according to the newly designed rule. The distance dierence needs to be (1) Variables initialize. calculated between the path with the largest flow and the path with the second largest flow. If the (2) According to the Formulae (1)–(3), the distance L and the flow Q are calculated. ij ij distance dierence is large, and the distance with the largest flow is shorter, the point with the largest (3) According to (2), we are going to complete the updating of conductivity and flow by Formula flow will be selected as the next point. Otherwise, we need to form a point set that includes the (5) and (3). Then, the final flow values are obtained by iterating until the condition (6) of termination shortest path point, the shorter path point, and the path point with the larger flow. Then, a random is satisfied. factor is added to achieve the selection of the next point when we select the points in point set. (4) According to the two indexes of flow and distance, the point is selected in a two-way parallel Finally, the optimized result of TPSMA will be obtained by iterating. method, and the specific contents are as follows: According to the principle of TPSMA on flow and distance, the algorithm is applied to the solution of TSP. As illustrated in Figure 4, the specific steps are as follows: Q Q (i) We select the of the largest flow value and the of the second largest flow value ibest ibetter (1) Variables initialize. from the points to be selected and define the point of P and P . At the same time, the shortest Qibest Qibetter (2) According to the Formulae (1)–(3), the distance L and the flow Q are calculated. ij ij L L distance and the second shortest distance are respectively found out and defined the ibest ibetter (3) According to (2), we are going to complete the updating of conductivity and flow by Formula (5) P P P P and point (3). of Then, the and final flow . As values follow are s, obtained P , by iterating and until ar the e tcondition aken and f (6) orm ofed a termination set P . Libest Libetter Qibetter Libest Libetter QL is satisfied. P is defined as the other choice of next point. The formula of point set is: Qibest (4) According to the two indexes of flow and distance, the point is selected in a two-way parallel PP = ,,P P method, and the specific contents are as follows: { } (9) QL Qibetter Libest Libetter (i) We select the Q of the largest flow value and the Q of the second largest flow value ibest ibetter And the optional point is: from the points to be selected and define the point of P and P . At the same time, the shortest Qibest Qibetter distance L and the second shortest distance L are respectively found out and defined the point ibest ibetter PP = { } (10) Qibest Qibest of P and P . As follows, P , P and P are taken and formed a set P . P is QL Libest Libetter Qibetter Libest Libetter Qibest e L L defined (ii)as the is de other fin choice ed as the of next differ point. ence be The tween formula of point and set is: . If the distance of the path with LQ Qibest Qibetter n o the largest flow is too long or similar to others, the point in P will be selected as the next point by QL P = P , P , P (9) QL Qibetter Libest Libetter random factors. Otherwise, P , which is obtained by the maximum flow, is going to be chosen as Qibest the next point. The improved rule, which is a two-way parallel selective method of flow and distance, And the optional point is: n o is formulated by P , as expressed in (11) P and (1 =2)P : (10) Qibest Qibest next (ii) e is defined as the dierence between L and L . If the distance of the path with LQ eL=− QibestL Qibetter (11) LQ Qibest Qibetter the largest flow is too long or similar to others, the point in P will be selected as the next point by QL Rand P , e > ε random factors. Otherwise, P , which is obtained by the maximum flow, is going to be chosen as () Qibest QL LQ P = (12) next Pe , ≤ ε Qibest LQ “Pipeline Culture”model of slime mold Appl. Sci. 2020, 10, 6180 6 of 15 Appl. Sci. 2020, 10, x FOR PEER REVIEW 6 of 20 where L is the distance of the path with the maximum flow, L is the distance of the path Qibest Qibetter the next point. The improved rule, which is a two-way parallel selective method of flow and distance, with the second largest flow, represents the next path point to be selected, Rand P () next is formulated by P , as expressed in (11) and (12): QL next represents a random value in P , which is the set of selected points, and ε is the determining QL e = L L (11) LQ Qibest Qibetter parameter of distance difference. By adjusting the value of ε , which is usually a negative, an appropriate cut-off point will be obtained. Rand P , e > " QL LQ P = (12) next > (5) According to (4), a circulation will be completed when all the points are selected and the P , e " LQ Qibest optimization result has been obtained. Then, the iterations are carried out until getting the where L is the distance of the path with the maximum flow, L is the distance of the path Qibest Qibetter termination condition. L , which is the set including all the paths, is obtained. Finally, the formula ength with the second largest flow, P represents the next path point to be selected, Rand P represents a next QL that can get the optimal path L in L is obtained by: best ength random value in P , which is the set of selected points, and " is the determining parameter of distance QL dierence. By adjusting the value of ", which is usually a negative, an appropriate cut-o point will LL = min { } (13) best ength be obtained. Figure Figure 4. 4. Flow Flow chart chart of T of TPSMA. PSMA. According to the above-mentioned steps, TPSMA is summarized as Table 1: Appl. Sci. 2020, 10, 6180 7 of 15 (5) According to (4), a circulation will be completed when all the points are selected and the optimization result has been obtained. Then, the iterations are carried out until getting the termination condition. L , which is the set including all the paths, is obtained. Finally, the formula that can get ength the optimal path L in L is obtained by: best ength n o L = min L (13) best ength According to the above-mentioned steps, TPSMA is summarized as Table 1: Table 1. The steps of TPSMA. A Two-Way Parallel Slime Mold Algorithm by Distance and Flow input: TSP path points output: TSP shortest path (a) Initialization process Step 1 Initialize variables and parameters (b) Calculate the distance and flow Step 2 Get the distance L according to Formula (1) ij Step 3 Get the flow Q by the Formulas (2) and (3) of Kirchho’s Laws ij Step 4 Update the conductivity by Formula (5) Step 5 Return to Step 3 to cycle until iterative condition is terminated and obtain the stable flow value Q ij (c) The point selection by a two-way parallel method Step 6 Select the points P , P , P and L , and gain Libest Libetter Qibest Qibetter two sets of points according to the Formula (9) and (10) Step 7 Obtain the value e by subtracting L from L . LQ Qibest Qibetter Then, complete the selection of point in turn by using the two-way parallel Formula (11) and (12) Step 8 Return to Step 6 to finish the iterations until the iterative condition is satisfied, then get all the paths L ength (d) Obtain the result Step 9 Obtain and output the optimal path L by the Formula (13) best 4. Simulation and Analysis of Results Traveling Salesman Problem Library (TSPLIB) is a library of sample instances for the TSP from various sources. Each set is a two-dimensional array containing horizontal and vertical coordinates of some cities, which is used as a test of intelligent algorithms. By using TSPLIB data to test and simulate in MATLAB, the experimental results were compared and analyzed to verify the eectiveness of TPSMA. 4.1. Result of Simulation We selected four datasets of TSP in TSPLIB; ulysses16, city31, eil51, gr96, and bier127. Of these, city31 is the data set of longitude and latitude coordinates of the locations of 31 cities in China. Multiple sets of data are selected to ensure the reliability of the conclusions. At the same time, the number of the four groups of datasets is 16, 31, 51, 96, and 127, separately and increasing in order. It increases the diculty of optimization and fully verifies the performance of the algorithm. Figures 5–9 are the simulation results obtained about five groups of TSP data by SMA and TPSMA. It can be seen from the figures that the results under the five groups of data are obviously improved by TPSMA. Appl. Sci. 2020, 10, x FOR PEER REVIEW 8 of 20 Appl. Sci. 2020, 10, 6180 8 of 15 Appl. Sci. 2020, 10, x FOR PEER REVIEW 9 of 20 (a) The result by SMA (b) The result by TPSMA Figure 5. Figure 5. The s The simulation imulation of u of ulysses16. lysses16. Appl. Sci. 2020, 10, x FOR PEER REVIEW 10 of 20 (a) The result by SMA (b) The result by TPSMA Figure 6. The simulation of city31. Figure 6. The simulation of city31. Appl. Sci. 2020, 10, x FOR PEER REVIEW 11 of 20 (a) The result by SMA (b) The result by TPSMA Figure 7. Figure 7. The simulation of e The simulation of eil51. il51. (a) The result by SMA (b) The result by TPSMA Figure Figure 8. 8. The simulation of gr The simulation of gr96. 96. Appl. Sci. 2020, 10, x FOR PEER REVIEW 12 of 20 Appl. Sci. 2020, 10, 6180 9 of 15 (a) The result by SMA (b) The result by TPSMA Figure 9. Figure 9. The simulation of b The simulation of bier127. ier127. 4.2. Analysis of Selection Processing 4.2. Analysis of Selection Processing Taking ulysses16 data as an example, the optimization ability of TPSMA is verified by analyzing Taking ulysses16 data as an example, the optimization ability of TPSMA is verified by analyzing the optimization process. Figure 5 and Table 2 are simulations and partially obtained paths under the optimization process. Figure 5 and Table 2 are simulations and partially obtained paths under ulysses16 data. In the Tables, red represents the current point and optimal result, green represents the ulysses16 data. In the Tables, red represents the current point and optimal result, green represents selected point by SMA and TPSMA, and blue represents the remaining points in the set of candidate the selected point by SMA and TPSMA, and blue represents the remaining points in the set of points by TPSMA. Figure 10 shows the flow and distance values from path point 12 to the remaining candidate points by TPSMA. Figure 10 shows the flow and distance values from path point 12 to the path points, and their ranking of each point. remain Appl. Sci. ing 2020 pa, th poin 10, x FOR P ts, EER and R the EVIEW ir r anking of each point. 13 of 20 When the current point is 12, it can be found in Figure 10 that the points of the largest flow, the second largest flow, the shortest distance, and the second shortest distance in the remaining candidate points are 15, 16, 8, and 11, respectively. In Table 2, point 15 with the maximum flow is selected by SMA, and the path 5 is the optimization result of SMA with a path length of 103.174. There are four candidate points, 15, 16, 8, and 11, which are obtained by TPSMA. Moreover, the path 1 to path 4 in Table 2 can be realized by adding random factors, and path 1 is the optimal result of TPAMA with a path length of 77.8372. Table 2. Partial path optimization results under ulysses16 data. No. Path Optimization Results (Points Order ) Path Length 1 13 14 4 6 9 12 11 8 5 2 7 10 15 16 3 1 77.8372 2 13 14 4 9 6 12 8 11 5 2 10 15 7 1 16 3 88.8287 3 13 14 4 9 6 12 15 16 1 2 7 10 11 8 5 3 94.3883 4 13 14 4 6 9 12 16 15 7 10 11 8 5 2 3 1 88.2858 5 14 9 10 6 7 3 11 1 2 12 15 8 5 6 4 13 103.174 Figure 10. Data of flow and distance from point 12 to the remaining points, and the selection from 12 Figure 10. Data of flow and distance from point 12 to the remaining points, and the selection from 12 to the next point by SMA and TPSMA. to the next point by SMA and TPSMA. When the current point is point 3, it can be obtained in Figure 11 that the candidate points are 1, 2, and 7. Path 1 to path 5 in Table 3 can be obtained by random factors, and path 1 is the final result of TPSMA in this paper. Point 1 is selected after point 3 and the path length is 77.8372 in path 1. Furthermore, Table 3 shows that the next point of point 3 in path 2 is point 1 and the next choice of point 12 is point 11. The choices of two points is the same as the result of TPSMA in path 2. In contrast, only the next point of point 3 is the same as path 1 in path 3 and the next point of point 3 and point 12 in path 4 and path 5 is different from path 1. What is more, the results show that the path lengths of path 1, path 2, and path 3 are better than path 4 and path 5. From Table 3, it can be analyzed that the more points that selected by distance and flow, the better the result will be obtained. The above results show that TPSMA of two parameters with flow and distance is superior to SMA. Table 3. Partial path optimization results under ulysses16 data. No. Path Optimization Results (Points Order ) Path Length 1 16 3 1 13 14 4 6 9 12 11 8 5 2 7 10 15 77.8372 2 16 3 1 2 14 13 4 9 6 12 11 8 15 10 5 7 80.9361 3 16 3 1 2 4 1 6 9 14 12 15 5 8 11 10 7 86.9835 4 16 3 2 1 4 13 6 9 14 12 15 11 5 7 8 10 98.9656 5 16 3 7 1 11 5 8 10 2 12 15 4 9 6 14 13 103.619 Appl. Sci. 2020, 10, 6180 10 of 15 Table 2. Partial path optimization results under ulysses16 data. No. Path Optimization Results (Points Order) Path Length 1 13 14 4 6 9 12 11 8 5 2 7 10 15 16 3 1 77.8372 2 13 14 4 9 6 12 8 11 5 2 10 15 7 1 16 3 88.8287 3 13 14 4 9 6 12 15 16 1 2 7 10 11 8 5 3 94.3883 4 13 14 4 6 9 12 16 15 7 10 11 8 5 2 3 1 88.2858 5 14 9 10 6 7 3 11 1 2 12 15 8 5 6 4 13 103.174 When the current point is 12, it can be found in Figure 10 that the points of the largest flow, the second largest flow, the shortest distance, and the second shortest distance in the remaining candidate points are 15, 16, 8, and 11, respectively. In Table 2, point 15 with the maximum flow is selected by SMA, and the path 5 is the optimization result of SMA with a path length of 103.174. There are four candidate points, 15, 16, 8, and 11, which are obtained by TPSMA. Moreover, the path 1 to path 4 in Table 2 can be realized by adding random factors, and path 1 is the optimal result of TPAMA with a path length of 77.8372. When the current point is point 3, it can be obtained in Figure 11 that the candidate points are 1, 2, and 7. Path 1 to path 5 in Table 3 can be obtained by random factors, and path 1 is the final result of TPSMA in this paper. Point 1 is selected after point 3 and the path length is 77.8372 in path 1. Furthermore, Table 3 shows that the next point of point 3 in path 2 is point 1 and the next choice of point 12 is point 11. The choices of two points is the same as the result of TPSMA in path 2. In contrast, only the next point of point 3 is the same as path 1 in path 3 and the next point of point 3 and point 12 in path 4 and path 5 is dierent from path 1. What is more, the results show that the path lengths of path 1, path 2, and path 3 are better than path 4 and path 5. From Table 3, it can be analyzed that the more points that selected by distance and flow, the better the result will be obtained. The above results show that TPSMA of two parameters with flow and distance is superior to SMA. Appl. Sci. 2020, 10, x FOR PEER REVIEW 14 of 20 Q4 D15 Q5 D13 4 Q9 D12 Q7 D11 Q13 D14 Q14 D10 Q3 D6 Q6 D9 Q8 D7 Q10 D8 Q11 D3 1 Q2 D2 Q12 D4 D5 Q15 Q1 D1 Figure 11. Data of flow and distance from point 3 to the remaining points. Figure 11. Data of flow and distance from point 3 to the remaining points. Table 3. Partial path optimization results under ulysses16 data. 4.3. Analysis of Diversity No. Path Optimization Results (Points Order) Path Length Under the data of ulysses16, as shown in Figure 12, SMA only gets path 1. However, the 1 16 3 1 13 14 4 6 9 12 11 8 5 2 7 10 15 77.8372 optimization process of TPSMA obtains the path 1 and path 2, which is the same as the length result 2 16 3 1 2 14 13 4 9 6 12 11 8 15 10 5 7 80.9361 of SMA. At the same time, in Figure 12, TPSMA gets two groups of paths 5 to 6 and paths 7 to 9. The 3 16 3 1 2 4 1 6 9 14 12 15 5 8 11 10 7 86.9835 path lengths of each group are the same but the paths have different orders of points. Similarly, 4 16 3 2 1 4 13 6 9 14 12 15 11 5 7 8 10 98.9656 TPSMA will produce a variety of cases due to the addition of random factors. Therefore, the diversity 5 16 3 7 1 11 5 8 10 2 12 15 4 9 6 14 13 103.619 and comprehensiveness of the optimization results are increased. Appl. Sci. 2020, 10, 6180 11 of 15 4.3. Analysis of Diversity Under the data of ulysses16, as shown in Figure 12, SMA only gets path 1. However, the optimization process of TPSMA obtains the path 1 and path 2, which is the same as the length result of SMA. At the same time, in Figure 12, TPSMA gets two groups of paths 5 to 6 and paths 7 to 9. The path lengths of each group are the same but the paths have dierent orders of points. Similarly, TPSMA will produce a variety of cases due to the addition of random factors. Therefore, the diversity and comprehensiveness of the optimization results are increased. Appl. Sci. 2020, 10, x FOR PEER REVIEW 15 of 20 Figure 12. Partial paths of TPSMA under ulysses16 data. Figure 12. Partial paths of TPSMA under ulysses16 data. 4.4. Analysis of Optimization Ability in Dierent Numbers of Points 4.4. Analysis of Optimization Ability in Different Numbers of Points Based on four sets of data from ulysses16, city31, eil51, gr96, and bier127, the simulation diagrams Based on four sets of data from ulysses16, city31, eil51, gr96, and bier127, the simulation are shown in Figures 5–9, and the results of data are shown in Table 4. Based on the analysis of the diagrams are shown in Figures 5–9, and the results of data are shown in Table 4. Based on the analysis data, it can be obtained that: of the data, it can be obtained that: Firstly, under the four sets of data, compared with SMA, the optimization ability of TPSMA is Table 4. Experimental data’s results of ulysses16, city31, eil51, gr96 and bier127. improved by 24.56, 36.10, 41.88, 49.83, and 52.93%, respectively. Therefore, the result of TPSMA is TSP Data Results of SMA Results of TPSMA Improved Percentages obviously better than SMA in Figure 13a. Secondly, as per the experimental results also shown in Figure 13b, the improved percentages ulysses16 103.1746 77.8372 24.56% city31 27,073 17,300 36.10% of TPSMA are gradually increased with the increasing number of points. It can be seen that TPSMA eil51 798.9 464.3 41.88% designed in this paper is more powerful in solving TSP with large points and complex distribution. gr96 1178 591 49.83% Thus, the experimental results fully verify the rationality and superiority of TPSMA. bier127 274,870 129,390 52.93% Table 4. Experimental data’s results of ulysses16, city31, eil51, gr96 and bier127. Firstly, under the four sets of data, compared with SMA, the optimization ability of TPSMA is TSP Data Results of SMA Results of TPSMA Improved Percentages improved by 24.56, 36.10, 41.88, 49.83, and 52.93%, respectively. Therefore, the result of TPSMA is ulysses16 103.1746 77.8372 24.56 % obviously better than SMA in Figure 13a. city31 27,073 17,300 36.10 % eil51 798.9 464.3 41.88 % gr96 1178 591 49.83 % bier127 274,870 129,390 52.93% Appl. Sci. 2020, 10, x FOR PEER REVIEW 16 of 20 Appl. Sci. 2020, 10, 6180 12 of 15 Comparison results Improved percentage 1.2 52.93% 49.83% 0.8 41.88% 0.6 36.10% 0.4 0.2 24.56% ulysses16 city31 eil51 gr96 bier127 ulysses16 city31 eil51 gr96 bier127 Improved percentages Slime mold algorithm Improved algorithm (a) Comparison results of SMA and TPSMA (b) Improved percentages by TPSMA Figure 13. Analysis results of SMA and TPSMA under the four sets of data. Figure 13. Analysis results of SMA and TPSMA under the four sets of data. 4.5. CSecondly omparisio,n wi as per th Ot the herexperimental Algorithms results also shown in Figure 13b, the improved percentages of TPSMA are gradually increased with the increasing number of points. It can be seen that TPSMA Based on some sets that have large points in TSPLIB, we can obtain the following by experiment. designed in this paper is more powerful in solving TSP with large points and complex distribution. Table 5 is the results of each heuristic and bionic algorithm under different data sets. From the Thus, the experimental results fully verify the rationality and superiority of TPSMA. experimental results in Figure 14, the TPSMA results are obviously better than GA and PSO, and similar to ACO. Since the initial pheromone distribution of ACO is unpredictable, the reasonable 4.5. Comparision with Other Algorithms distribution of pheromones needs to be gradually formed by iteration. Therefore, the convergence Based on some sets that have large points in TSPLIB, we can obtain the following by experiment. speed is very slow. Although TPSMA optimization results were slightly worse than ACO, TPSMA Table 5 is the results of each heuristic and bionic algorithm under dierent data sets. From the optimization speed was significantly better than ACO. Compared with the current best optimization experimental results in Figure 14, the TPSMA results are obviously better than GA and PSO, and similar result, which is from LKH, the result of TPSMA is not as good as LKH; however, it is closer to the to ACO. Since the initial pheromone distribution of ACO is unpredictable, the reasonable distribution optimal result than other algorithms. Moreover, LKH is in virtue of the 5-opt principle which is of pheromones needs to be gradually formed by iteration. Therefore, the convergence speed is very based on the λ-opt algorithm. The more task points there are, the more iteration time it will cost, and slow. Although TPSMA optimization results were slightly worse than ACO, TPSMA optimization the convergence performance will be poor. speed was significantly better than ACO. Compared with the current best optimization result, which is We can get the degree of difficulty about algorithm principle and the algorithm time complexity from LKH, the result of TPSMA is not as good as LKH; however, it is closer to the optimal result than in Table 5. It can be seen that the TPSMA algorithm is simple in principle and easy to analyze. At the other algorithms. Moreover, LKH is in virtue of the 5 opt principle which is based on the opt 3 4 same time, the algorithm time complexity of TPSMA ( N ) is less 1/N times than that of ACO ( N ), algorithm. The more task points there are, the more iteration time it will cost, and the convergence 2 5 and less times than that of LKH ( ). TPSMA has fast convergence speed and short 1/N N performance will be poor. optimization time due to low algorithm time complexity, especially in large data points. Table 5. Comparisons of optimization results and algorithm features. Table 5. Comparisons of optimization results and algorithm features. Name PSO GA ACO LKH TPSMA Name PSO GA ACO LKH TPSMA eil51 1257 519 453 426 464 eil51 1257 519 453 426 464 eil76 2040 727 583 538 620 eil76 2040 727 583 538 620 lin105 96,429 30,167 15,303 14,379 16,424 Results of path length bier127 542,558 196,276 128,147 118,282 129,390 lin105 96,429 30,167 15,303 14,379 16,424 Results of path length kroA200 290,368 87,786 33,471 29,368 34,972 bier127 542,558 196,276 128,147 118,282 129,390 gil262 23,780 7769 2779 2378 2881 kroA200 290,368 87,786 33,471 29,368 34,972 3 3 5 3 Algorithm Time Complexity N N N N N gil262 23,780 7769 2779 2378 2881 Theoretical diculty Complexity Simple Complexity Complexity Simple 3 3 4 5 Algorithm Time Complexity N N N N Theoretical difficulty Complexity Simple Complexity Complexity Simple Proption Percentage Appl. Sci. 2020, 10, 6180 13 of 15 Appl. Sci. 2020, 10, x FOR PEER REVIEW 17 of 20 2040 96429 542558 290368 23780 727 33471 eil51 eil76 lin105 bier127 kroA200 gil262 Category of data set PSO GA ACO LKH TPSMA Figure 14. Comparison of optimization results under TSPLIB data sets. Figure 14. Comparison of optimization results under TSPLIB data sets. We can get the degree of diculty about algorithm principle and the algorithm time complexity According to the above analysis, it can be concluded from Figure 15 that TPSMA has good in Table 5. It can be seen that the TPSMA algorithm is simple in principle and easy to analyze. At the 3 4 searching ability by a unique way, and fast optimization speed by low time complexity. At the same same time, the algorithm time complexity of TPSMA (N ) is less 1/N times than that of ACO (N ), 2 5 time, TPSMA is simple in theory and easy to research, therefore it is better to study than the others. and less 1/N times than that of LKH (N ). TPSMA has fast convergence speed and short optimization Furthermore, the research time of TPSMA is short and the algorithm is not mature enough—there time due to low algorithm time complexity, especially in large data points. are many performances and potential to be developed due to its unique optimization method and According to the above analysis, it can be concluded from Figure 15 that TPSMA has good effectiveness. searching ability by a unique way, and fast optimization speed by low time complexity. At the same time, TPSMA is simple in theory and easy to research, therefore it is better to study than Appl. the Sci. others. 2020, 10 Furthermor , x FOR PEER R e,EVIEW the r esearch time of TPSMA is short and the algorithm is not 18 of matur 20 e enough—there are many performances and potential to be developed due to its unique optimization method and eectiveness. Time Optimization Degree of Potential of Features of Complexity ability difficulty Research Algorithm ++ + GA PSO ACO LKH TPSMA Figure 15. Figure 15.Comparison and Comparison andanalysis o analysis of f alg algorithm orithm characteristi characteristics. cs. 5. Summary In this paper, the two-way parallel selection principle of distance and flow is adopted by TPSMA, and random factor is added to improve optimization ability and diversity. Through the experimental results of TSPLIB data, the path length that is obtained by TPSMA is obviously reduced. What is more, the optimization ability is gradually enhanced with the increase in the number of path points. At the same time, TPSMA can get all the paths that meet the requirements to realize the diversity of path results. The above results prove the feasibility and superiority of TPSMA in solving TSP. The proposed method will show partial reversal paths and diagonal paths in the searching process, which could have an impact on the search results. Follow-up research can start with the direction of flow, to research and improve the algorithm performance. Author Contributions: Conceptualization, M.L., Y.L. and Q.H.; methodology, M.L. and Y.L.; software A.L.; validation, Y.L.; formal analysis, M.L. and Y.L.; investigation, Q.H. and M.X.; resources, M.Z. and L.C.; data curation, Q.H.; writing—original draft preparation, M.L.; writing—review and editing, M.L., M.Z. and N.Q.; visualization, M.X.; supervision, L.C.; project administration, N.Q.; funding acquisition, M.Z. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the National Natural Science Foundation of China (No. 11672290), Science and Technology Development Plan of Jilin province (No. 2018020102GX), and Jilin Province and the Chinese Academy of Sciences cooperation in science and technology high-tech industrialization special funds project (No. 2018SYHZ0004). Conflicts of Interest: The authors declare no conflict of interest. We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted. References 1. Lu, Y.; Liu, Y.; Gao, C.; Tao, L.; Zhang, Z. A Novel Physarum-Based Ant Colony System for Solving the Real-World Traveling Salesman Problem. In Proceedings of the International Conference in Swarm Intelligence, Hefei, China, 17–20 October 2014; Volume 8794, pp. 173–180. 2. Ouaarab, A.; Ahiod, B.; Yang, X.S. Random-key cuckoo search for the travelling salesman problem. Soft Comput. 2015, 19, 1099–1106. 3. Ahmadi, E.; Goldengorin, B.; Gürsel, S.; Mosadegh, H. A hybrid method of 2-TSP and novel learning-based GA for job sequencing and tool switching problem. Appl. Soft Comput. 2018, 65, 214–229. Relative path length Appl. Sci. 2020, 10, 6180 14 of 15 5. Summary In this paper, the two-way parallel selection principle of distance and flow is adopted by TPSMA, and random factor is added to improve optimization ability and diversity. Through the experimental results of TSPLIB data, the path length that is obtained by TPSMA is obviously reduced. What is more, the optimization ability is gradually enhanced with the increase in the number of path points. At the same time, TPSMA can get all the paths that meet the requirements to realize the diversity of path results. The above results prove the feasibility and superiority of TPSMA in solving TSP. The proposed method will show partial reversal paths and diagonal paths in the searching process, which could have an impact on the search results. Follow-up research can start with the direction of flow, to research and improve the algorithm performance. Author Contributions: Conceptualization, M.L., Y.L. and Q.H.; methodology, M.L. and Y.L.; software A.L.; validation, Y.L.; formal analysis, M.L. and Y.L.; investigation, Q.H. and M.X.; resources, M.Z. and L.C.; data curation, Q.H.; writing—original draft preparation, M.L.; writing—review and editing, M.L., M.Z. and N.Q.; visualization, M.X.; supervision, L.C.; project administration, N.Q.; funding acquisition, M.Z. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the National Natural Science Foundation of China (No. 11672290), Science and Technology Development Plan of Jilin province (No. 2018020102GX), and Jilin Province and the Chinese Academy of Sciences cooperation in science and technology high-tech industrialization special funds project (No. 2018SYHZ0004). 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